Anti-covering Problems - Location Science - page 114

Daskin et al. ( 1989 ), Kim and Murray ( 2008 ), Murray ( 2005 ), Murray and Wei
( 2013 ), Tong ( 2012 ), Tong and Murray ( 2009 ). For this reason, some papers are
found in which the regional aspect is directly handled. See for instance Blanquero
and Carrizosa ( 2013 ), Carrizosa et al. ( 1995 , 1998c ), Fekete et al. ( 2005 ), Yao and
Murray ( 2014 ) for single-facility Weber problems with regional demand, Murat
et al. ( 2010 ) for a heuristic method for the extension to p facilities, and Tong ( 2012 ),
Tong and Murray ( 2009 ) for discrete covering problems, in which the individuals
are identified with objects (polygons) in the plane, which can be considered as fully
or partially covered.
The remainder of the chapter is structured as follows. In Sect. 6.2 , a rather general
p-facility covering model for continuously distributed demand is described; how
to address the optimization problem is presented in Sect. 6.3 , and illustrated in
Sect. 6.4 . Conclusions and future lines of research are outlined in Sect. 6.5 .

6.2 Regional Covering Model

Location models are specific in the way the interactions are modeled. Two types of
interactions take place, namely, individual-facility interactions and facility-facility
interactions. Depending on the specific problem, just one or the two types of
interactions may be relevant; see e.g. Erkut and Neuman ( 1989 ).
Since these two types of interactions have different nature, they are discussed
separately in what follows.

6.2.1 Individual-Facility Interactions

For a given individual location a and any facility location x; let c.a;x/ 2 Œ0;1
denote how much a is covered (affected) by the facility at x: In its general form,
c. ; / may be any function ' W R C ! Œ0;1; which is non-increasing in the

(Euclidean) distance k x a k separating a and x;

c.a;x/ D '. k x a k /; (6.1)

so that, the lower the distance, the higher the coverage. This assumption, yet
sensible, may not be sound for specific problems of locating undesirable facilities;
for instance, Karkazis and Papadimitriou ( 1992 ) addresses the problem of locating
a polluting plant whose pollutant is discharged by means of high stacks, and thus
maximal interaction (damage) takes place at a non-negligible distance of the facility.
We remark that we are using the Euclidean distance, but this is not the only
choice of distance function kk found in the literature in covering models: see e.g.
Fernández et al. ( 2000 ) for a proposal of (weighted) ` p norms and Plastria ( 2002 )
for a thorough discussion on planar distances.